Scott's right... you can't really choose your own IP's unless they belong to RFC1918 space. At least not if you don't want to have any conflicts with other sites.
So let me see if I can answer some of your other questions...
1. Cables. If you connect similar devices then you need a cross-over cable. i.e. Router to Router, Switch to Switch, PC to PC then you'll need a cross-over cable. If you connect a Switch to Router, a PC to Switch then you need a regular ethernet cable. That's the general rule and there are always exceptions of course.
2. You said that you're going to be running IPCOP. That requires a dedicated PC with two network cards. I'm assuming that in your diagram the RED is the external interface of your IPCOP and GREEN is the internal interface. So in this case I would let the router give out a DHCP address to your IPCOP machine on the RED interface. Then you'll need to configure IPCOP to give out an IP address on the GREEN interface. You also need to disable any DHCP on your Switch (if it does more than just switch). Your IPCOP will also need to be configured to do NAT'ing and of course your Router will be doing NAT'ing as well.
Now, you can use the same IP subnet on the RED network and the GREEN network and that will work because of the double NATing but I don't recommend you do that since a) it could get confusing if you're new and b) there's plenty of other IP space. If the router uses 192.168.0.0/16 then choose something else in either the 10.0.0.0/8 or 172.16.0.0/12 ranges.
If you want to assign IP's manually then you'll need to make sure of the following:
- Your router is setup to NAT on the RED side
-- You can route through it on the RED side
-- Suggest you use 192.168.1.0/24 subnet
--- 192.168.1.1/24 for your router (this is the GW for your IPCOP RED side)
--- 192.168.1.2/24 for your RED interface on your IPCOP
- Your IPCOP is setup to NAT on the GREEN side
-- You can route through it on the GREEN side
-- Suggest you use 10.0.0.0/24 subnet
--- 10.0.0.1/24 for your GREEN interface on your IPCOP (this is your PC's GW address also)
--- 10.0.0.x/24 for your internal hosts
As you can see static is more complex but doable if that's what you really want to do.
Now having said all of that. Why do you want to use your Router AND IPCOP? I would personally use only one of them (I'd use the IPCOP).
Hope that helps.
Casey